Montel's Theorem and subspaces of distributions which are $\Delta^m$-invariant

Abstract

We study the finite dimensional spaces $V$ which are invariant under the action of the finite differences operator $\Delta_h^m$. Concretely, we prove that if $V$ is such an space, there exists a finite dimensional translation invariant space $W$ such that $V\subseteq W$. In particular, all elements of $V$ are exponential polynomials. Furthermore, $V$ admits a decomposition $V=P\oplus E$ with $P$ a space of polynomials and $E$ a translation invariant space. As a consequence of this study, we prove a generalization of a famous result by P. Montel which states that, if $f:\mathbb{R}\to \mathbb{C}$ is a continuous function satisfying $\Delta_{h_1}^mf(t) = \Delta_{h_2}^mf(t)=0$ for all $t\in\mathbb{R}$ and certain $h_1,h_2\in\mathbb{R}\setminus\{0\}$ such that $h_1/h_2\not\in\mathbb{Q}$, then $f(t)=a_0+a_1t+\cdots+a_{m-1}t^{m-1}$ for all $t\in\mathbb{R}$ and certain complex numbers $a_0,a_1,\cdots,a_{m-1}$. We demonstrate, with quite different arguments, the same result not only for ordinary functions $f(t)$ but also for complex valued distributions. Finally, we also consider in this paper the subspaces $V$ which are $\Delta_{h_1h_2\cdots h_m}$-invariant for all $h_1,\cdots,h_m\in\mathbb{R}$.